scholarly journals Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant

2017 ◽  
Vol 53 (1) ◽  
pp. 121-150 ◽  
Author(s):  
Xuezhang Chen ◽  
Pak Tung Ho ◽  
Liming Sun
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Compact Manifolds ◽  
Manifolds With Boundary ◽  
Conformal Invariant

4-Dimensional manifolds with nonnegative scalar curvature and CMC boundary

2020 ◽  
pp. 1950094
Author(s):  
Yaohua Wang
Keyword(s):  
Scalar Curvature ◽  
Upper Bound ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Outer Boundary ◽  
Compact Manifolds ◽  
Manifolds With Boundary ◽  
Flat Extension ◽  
Asymptotically Flat ◽  
3 Dimensional

In this paper, we will consider 4-dimensional manifolds with nonnegative scalar curvature and constant mean curvature (CMC) boundary. For compact manifolds with boundary, the influence of the nonnegativity of the region scalar curvature to the geometry of the boundary is considered. Some inequalities are established for manifolds with inner boundary and outer boundary. Even for compact manifolds without inner boundary, we can obtain some inequalities involving the geometric quantities of the boundary and give some obstruction. We also discuss the 4-dimensional asymptotically flat extension of the 3-dimensional Bartnik data with CMC boundary and provide the upper bound of the Bartnik mass.

scholarly journals About the Uniqueness of Conformal Metrics with Prescribed Scalar and Mean Curvatures on Compact Manifolds with Boundary

2011 ◽  
Vol 13 ◽  
pp. 71-79
Author(s):  
Gonzalo García ◽  
Jhovanny Muñoz
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Manifolds With Boundary ◽  
Conformal Metrics ◽  
Dirichlet Eigenvalue ◽  
Uniqueness Results ◽  
Manifold With Boundary ◽  
First Dirichlet Eigenvalue ◽  
Neumann Eigenvalue ◽  
Linear Elliptic Problem

Let (Mn, g) be an n—dimensional compact Riemannian manifold with boundary with n > 2. In this paper we study the uniqueness of metrics in the conformai class of the metric g having the same scalar curvature in M, dM, and the same mean curvature on the boundary of M, dM. We prove the equivalence of some uniqueness results replacing the hypothesis on the first Neumann eigenvalue of a linear elliptic problem associated to the problem of conformai deformations of metrics for one about the first Dirichlet eigenvalue of that problem. Keywords: Conformal metrics, scalar curvature, mean curvature.

scholarly journals Compactness and non-compactness for the Yamabe problem on manifolds with boundary

2017 ◽  
Vol 2017 (724) ◽  
Author(s):  
Marcelo M. Disconzi ◽  
Marcus A. Khuri
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Scalar Curvature ◽  
Yamabe Problem ◽  
Riemannian Structure ◽  
Manifolds With Boundary ◽  
Conformal Deformation ◽  
Zero Mean Curvature

AbstractWe study the problem of conformal deformation of Riemannian structure to constant scalar curvature with zero mean curvature on the boundary. We prove compactness for the full set of solutions when the boundary is umbilic and the dimension

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